I'm currently using the free version and I'd like to upgrade to the Pro version (with MySQL). However, I'll put the upgraded version on a different domain. Should I just install the pro version and then replace the subscribers.txt file or ... What's the best way to do this?

That is probably your best option. You just place the existing subscribers.txt under the new list's directory (each list with Subscriber Pro has a separate directory). Then just run the upgrade.cgi script. There's nothing to it really, and the manual has a small bit about this process with a very basic example you can see at: http://npscripts.com/subscriber-pro/manual/upgrading.html.

If you then want to use a database backend, there is another script included that can convert the plain text file. But first, you have to set up the database options. This entire process is detailed on this page: http://npscripts.com/subscriber-pro/manual/sql.html.